The weak equivalences play the role of ‘homotopy equivalences’ or something a bit more general. Already in the case of topological spaces, it is useful to say that two spaces have the same homotopy type if there is a map from one to the other that induces isomorphisms on homotopy groups for any choice of basepoint in the first space. These maps are more general than homotopy equivalences, so they are called ‘weak equivalences’.

A bit more technically: we can define an (∞,1)-category starting from any category with weak equivalences. The idea is that this (∞,1)-category keeps track of objects in our original category, morphisms between objects, homotopies between maps, homotopies between homotopies, and so on, ad infinitum. However, the extra structure of a model category makes it easier to work with this (∞,1)-category. We can obtain this (∞,1)-category in various ways, including simplicial localization if we want to obtain a simplicially enriched category as a variant of (∞,1)-category. Alternatively, to obtain a quasicategory, given a model category MM, the simplicial nerve NΔ(Mcf)N_\Delta(M_{cf}) of the subcategory Mcf⊂MM_{cf}\subset M of cofibrant-fibrant objects is a quasicategory. We say this (∞,1)-category is presented (or modeled) by the model category, and that the objects of the model category are models for the objects of this (∞,1)(\infty,1)-category. Not every (∞,1)-category is obtained in this way (otherways it would necessarily have homotopy limits and colimits).

In this sense model categories are ‘models for homotopy theory’ or ‘categories of models for homotopy theory’. (The latter sense was the one intended by Quillen, but the former is also a useful way to think.)

Definition

A model structure on a categoryCC consists of three distinguished classes of morphisms - the cofibrationsCof⊂Mor(C)Cof \subset Mor(C), the fibrationsFibFib, and the weak equivalencesWW - satisfying the following two properties.

(i) WW makes CC into a category with weak equivalences, meaning that it is closed under 2-out-of-3: given a composable pair of morphisms f,gf,g, if two out of the three morphisms f,g,gff, g, g f are in WW, so is the third.

Often, the fibrant and cofibrant objects are the ones one is “really” interested in, but the category consisting only of these is not well-behaved (as a 1-category). The factorizations supply fibrant and cofibrant replacement functors which allow us to treat any object of the model category as a ‘model’ for its fibrant-cofibrant replacement.

Variants

Slight variations on the axioms

Quillen’s original definition required only finite limits and colimits, which are enough for the basic constructions. Colimits of larger cardinality are sometimes required for the small object argument, however.

Some authors, notably Mark Hovey, require that the factorizations given by (ii) are actually functorial. In practice, Quillen’s small object argument means that many model categories can be made to have functorial factorizations.

Enhancements of the axioms

There are several extra conditions that strengthen the notion of a model category:

Weaker axiom systems

A category of fibrant objects has a notion of just weak equivalences and fibrations, none of cofibrations. As the name implies, all of its objects are fibrant; the canonical example is the subcategory of fibrant objects in a model category.

A Waldhausen category dually has a notion of weak equivalences and cofibrations, and all of its objects are cofibrant.

Properties

Closure of morphism classes under retracts

As a consequence of the definition, the classes Cof,FibCof, Fib, and WW are all closed under retracts in the arrow categoryArrCArr C and under composition and contain the isomorphisms of CC.

This is least obvious in the case of WW. In the presence of functorial factorizations, it is easy to show that closure under retracts follows from axioms (i) and (ii); with a bit of cleverness, this can also be done without functoriality.

Redundancy in the defining factorization systems

It is clear that given a model category structure, any two of the three classes of special morphisms (cofibrations, fibrations, weak equivalences) determine the third:

given WW and CC, we have F=RLP(W∩C)F = RLP(W \cap C);

given WW and FF, we have C=LLP(W∩F)C = LLP(W \cap F);

given CC and FF, we find WW as the class of morphisms which factor into a morphism in C∩WC \cap W followed by a morphism in F∩WF \cap W.

But, in fact, already the cofibrations and the fibrant objects determine the model structure.

Proposition

A model structure (C,W,F)(C,W,F) on a category 𝒞\mathcal{C} is determined by its class of cofibrations and its class of fibrant objects.

Proof

So let ℰ\mathcal{E} with C,F,W⊂Mor(ℰ)C,F,W \subset Mor(\mathcal{E}) be a model category.

By the above remark it is sufficient to show that the cofibrations and the fibrant objects determine the class of weak equivalences. Moreover, these are already determined by the weak equivalences between cofibrant objects, because for u:A→Bu : A \to B any morphism, functorial cofibrant replacement ∅↪A^→≃A\emptyset \hookrightarrow \hat A \stackrel{\simeq}{\to} A and ∅↪B^→≃B\emptyset \hookrightarrow \hat B \stackrel{\simeq}{\to} B with 2-out-of-3 implies that uu is a weak equivalence precisely if u^:A^→B^\hat u : \hat A \to \hat B is.

By the nature of the homotopy categoryHoHo of ℰ\mathcal{E} and by the Yoneda lemma, a morphism u^:A^→B^\hat u : \hat A \to \hat B between cofibrant objects is a weak equivalence precisely if for every fibrant object XX the map

is an isomorphism, namely a bijection of sets. The equivalence relation that defines Ho(A^,X)Ho(\hat A,X) may be taken to be given by left homotopy induced by cylinder objects, which in turn are obtained by factoring codiagonals into cofibrations followed by acyclic fibrations. So all this is determined already by the class of cofibrations, and hence weak equivalences are determined by the cofibrations and the fibrant objects.

Categorical model structures

Of interest to category theorists is that many notions of higher categories come equipped with model structures, witnessing the fact that when retaining only invertible transfors between nn-categories they should form an (∞,1)(\infty,1)-category. Many of these are called

There is also another class of model structures on categorical structures, often called Thomason model structures (not to be confused with the notion of “Thomason model category”). In the “categorical” or “canonical” model structures, the weak equivalences are the categorical equivalences, but in the Thomason model structures, the weak equivalences are those that induce weak homotopy equivalences of nerves. Thomason model structures are known to exist on 1-categories and 2-categories, at least, and are generally Quillen equivalent to the Quillen model structures on topological spaces and simplicial sets (via the nerve construction).

Parametrized model structures

The parameterized version of the model structure on simplicial sets is a

and applying a general technique called Bousfield localization which forces a certain class of morphisms to become weak equivalences. It can also be thought of as forcing a certain class of objects to become fibrant.

There is an unpublished manuscript of Chris Reedy from around 1974 that’s been circulating as an increasingly faded photocopy. It’s been typed into LaTeX, and the author has given permission for it to be posted on the net: